Note : Les descriptions sont présentées dans la langue officielle dans laquelle elles ont été soumises.
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METHOD FOR PREDICTING FLUID FLOW
CROSS-REFERENCE TO RELATED APPLICATION
100011 This application claims the benefit of U.S. Provisional Patent
Application
61/170,896 filed 20 April 2009 entitled METHOD FOR PREDICTING FLUID FLOW.
TECHNICAL FIELD
100021 Aspects of methodologies and techniques disclosed herein relate to
methods for
efficiently and accurately solving water flow and sediment transport equations
numerically,
which may be used to predict fluid flow and optimize production of hydrocarbon
resources
from geologic formations based on such predictions.
BACKGROUND OF THE DISCLOSURE
100031 Numerical simulation is widely used in industrial fields as a method of
simulating
a physical system by using a computer. In most cases, there is a desire to
model the transport
processes occurring in the physical system. What is being transported is
typically mass,
energy, momentum, or some combination thereof By using numerical simulation,
it is
possible to model and observe a physical phenomenon and to determine design
parameters,
without actual laboratory experiments and field tests.
100041 The principle of numerical simulation is to numerically solve equations
describing
a physical phenomenon by a computer, such as fluid flow. Such equations are
generally
ordinary differential equations and partial differential equations. These
equations are
typically solved by linearizing the equations and using numerical methods such
as the finite
element method, the finite difference method, the finite volume method, and
the like. In each
of these methods, the physical system to be modeled is divided into smaller
gridcells or
blocks (a set of which is called a grid or mesh), and the state variables
continuously changing
in each gridcell are represented by sets of values for each gridcell. In the
finite difference
method, an original differential equation is replaced by a set of algebraic
equations to express
the fundamental principles of conservation of mass, energy, and/or momentum
within each
gridcell and transfer of mass, energy, and/or momentum transfer between
gridcells. These
equations can number in the millions. Such replacement of continuously
changing values by
a finite number of values for each gridcell is called "discretization." In
order to analyze a
phenomenon changing in time, it is necessary to calculate physical quantities
at discrete
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intervals of time called timesteps, irrespective of the continuously changing
conditions as a
function of time. Time-dependent modeling of the transport processes proceeds
in a sequence
of timesteps. Many numerical schemes have been used for solving the partial
differential
equation, including for example the well known Godunov scheme. In the Godunov
scheme,
conservative variables are considered as piecewise constant over the mesh
cells at each time
step and the time evolution is determined by the exact solution of what is
known as the
Riemann problem. To reduce computation time, the changes in the model
parameter values
defined for individual grid cells are typically piecewise-constant within the
cells rather than
continuously varying.
[0005] Godunov's scheme can be described as follows:
"In numerical analysis and computational fluid dynamics, Godunov's scheme is a
conservative numerical scheme, suggested by S. K. Godunov in 1959, for solving
partial
differential equations. In this method, the conservative variables are
considered as
piecewise constant over the mesh cells at each time step and the time
evolution is
determined by the exact solution of the Riemann problem (shock tube) at the
inter-cell
boundaries (Hirsch, C. (1990), Numerical Computation of Internal and External
Flows,
vol 2, Wiley.).
Following Hirsch, the scheme involves three distinct steps to obtain the
solution at t =
(n+1) At from the known solution at t = n At, as follows:
Step 1 Define piecewise constant approximation of the solution at t = (n+1)
At. Since the
piecewise constant approximation is an average of the solution over the cell
of size Ax ,
the spatial error is of order Ax , and hence the resulting scheme will be
first-order accurate
in space. Note that this approximation corresponds to a finite volume method
representation whereby the discrete values represent averages of the state
variables over
the cells. Exact relations for the averaged cell values can be obtained from
the integral
conservation laws.
Step 2 Obtain the solution for the local Riemann problem at the cell
interfaces. This is the
only physical step of the whole procedure. The discontinuities at the
interfaces are
resolved in a superposition of waves satisfying locally the conservation
equations. The
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original Godunov method is based upon the exact solution of the Riemann
problems.
However, approximate solutions can be applied as an alternative.
Step 3 Average the state variables after a time interval At. The state
variables obtained
after Step 2 are averaged over each cell defining a new piecewise constant
approximation
resulting from the wave propagation during the time interval At. To be
consistent, the
time interval At should be limited such that the waves emanating from an
interface do not
interact with waves created at the adjacent interfaces. Otherwise the
situation inside a cell
would be influenced by interacting Riemann problems. This leads to the CFL
condition
la.1 At < Ax /2 where lamaxl is the maximum wave speed obtained from the cell
eigenvalue(s) of the local Jacobian matrix.
The first and third steps are solely of a numerical nature and can be
considered as a
projection stage, independent of the second, physical step, the evolution
stage. Therefore,
they can be modified without influencing the physical input, for instance by
replacing the
piecewise constant approximation by a piecewise linear variation inside each
cell, leading
to the definition of second-order space-accurate schemes, such as the MUSCL
scheme."
[0006] In geologic modeling, a digital representation of the detailed
internal geometry
and rock properties of a subsurface earth volume, such as a petroleum
reservoir or a
sediment-filled basin is built. In the oil and gas industry, geologic models
provide geologic
input to reservoir performance simulations which are used to select locations
for new wells,
estimate hydrocarbon reserves, plan reservoir-development strategies, and/or
perform other
hydrocarbon extraction activities. The spatial distribution of permeability is
a key parameter
in characterizing reservoir performance, and, together with other rock and
fluid properties,
determines the producibility of the reservoir. For sandstone reservoirs, the
spatial distribution
of permeability is a function of the grain-size distribution of sands which
compose the
reservoir, the compartmentalization of those sands by barriers of finer
grained material, and
the mineralogy and burial history of the reservoir.
[0007] Most geologic models built for petroleum applications are in the
form of a three-
dimensional array of model blocks (cells), to which geologic and/or
geophysical properties
such as lithology, porosity, acoustic impedance, permeability, and water
saturation are
assigned (such properties will be referred to collectively herein as "rock
properties"). The
entire set of model blocks represents the subsurface earth volume of interest.
The goal of the
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geologic-modeling process is to assign rock properties to each model block in
the geologic
model.
[0008] The geologic modeling process can use many different data types,
including but
not limited to rock-property data obtained from cores, well logs, seismic
data, well test and
production data, and structural and stratigraphic surfaces that define
distinct zones within the
model. Other data types may include surface topographical data and aerial and
satellite
images that may be used to infer or predict subsurface geologic formations by
observing
and/or predicting similar geologic activities (erosion, deposition, fluid
flow, sediment flow,
etc.) on the earth's surface. In general, the resolution or spatial coverage
of the available data
is not adequate to uniquely determine the rock properties in every geologic
model cell. Other
assumptions about the variability in these properties are made in order to
populate all model
cells with reasonable property values. Geocellular techniques, object-based
modeling, and
process modeling are three main ways to populate the discretized geologic
volume with
properties.
[0009] In the geocellular approach, the relationship between properties in
nearby cells is
specified statistically. Geostatistical estimation methods (which may be
either deterministic
or probabilistic) are used to compute rock property values within cells. These
methods take
into account distance, direction, and spatial continuity of the rock property
being modeled.
Deterministic estimation methods commonly calculate a minimum-variance
estimate of the
rock property at each block. Probabilistic estimation methods develop
distributions of the
rock-property values and produce a suite of geologic model realizations for
the rock property
being modeled, with each realization theoretically being equally probable. The
spatial
continuity of a rock property may be captured by a variogram, a well-known
technique for
quantifying the variability of a rock property as a function of separation
distance and
direction. U.S. Pat. Nos. 5,838,634, 6,381,543 and 6,480,790 discuss
geocellular modeling
methods embodied in processing flows which include repetitive optimization
steps to drive
the geologic model toward conformance with geologic and geophysical data types
such as
wells, seismic surveys and subsurface fluid production and pressure data. Most
commercial
modeling software packages, including PETREL, GOCAD and STRATAMODEL, contain a
wide spectrum of geostatistical tools designed to fulfill the requirements of
reservoir
geologists and engineers. While these methods can readily accommodate data
control points
such as wells and geophysical constraints such as seismic data, they generally
do not closely
replicate the geologic structures observed in natural systems.
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[0010] In the object-based approach, subsurface reservoir volumes are
treated as
assemblages of geologic objects with pre-defined forms such as channels and
depositional
lobes. U.S. Pat. No. 6,044,328 discloses one object-based modeling scheme that
allows
geologists and reservoir engineers to select geologic objects from an analog
library to best
match the reservoir being modeled. The appropriateness of the analog is judged
by the
operator of the process based on their geologic experience. Most commercial
software
packages including PETREL, IRAP-RMS and GOCAD implement objects as volumetric
elements that mimic channels and lobes using simplified elements based on user
deformable
shapes such as half pipes and ellipses. Other examples of object-based models
are the model
depositional objects, such as a river belt and the braided stream network, are
placed
sequentially on top of each other according to some algorithms. While these
models try to
mimic the real depositional structures, they do not attempt to capture the
physics of water
flow and sediment transport that, over geologic time, determined the rock
properties at a
particular subsurface location.
[0011] Process-based models attempt to reproduce subsurface stratigraphy by
building
sedimentary deposits in chronological order relying to varying degrees on a
model or
approximation of the physical processes shaping the geology. Although process
models are
rarely used in current industrial practice, U.S. Pat. Nos. 5,844,799,
6,205,402 and 6,246,963
describe three such methods. These methods employ diffusion or rule-based
process models
to create basin-scale models with limited spatial detail inadequate for
reservoir performance
simulation. Another type of model is hydrodynamics-based gridding, which in
one aspect
may generate grid surfaces conforming to time surfaces during the process of
sedimentation.
Hydrodynamics-based gridding is the subject of U.S. Patent Application No.
11/629,822,
"Method for Geologic Modeling Through Hydrodynamics-Based Gridding (Hydro-
Grids)",
having inventors Li et al. and filed December 15, 2006.
[0012] Other types of process-based models that could potentially
provide the most
accurate representation of geologic structures are physics-based, process-
based models.
These models are based on basic physics of fluid flow and sediment transport,
and build the
subsurface stratigraphy by explicitly modeling the fluid flow, sediment
transport, erosion and
deposition that occurred during the formation of the reservoir. Compared with
other types of
geologic models, these physics-based process-based models can better preserve
the spatial
continuities of sedimentary body geometries as well as flow baffles and
barriers such as shale
and mud deposits.
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[0013] A component in the physics based process-based model is to solve
the fluid flow
and sediment transport equations. Often this involves solving shallow water or
depth-
averaged turbidity current equations. A problem in solving shallow-water and
depth-averaged
turbidity current equations using Godunov-type schemes is the imbalance
between flux
gradients and bed slope source terms due to the discretized representations of
flow and the
underlying topography. For stationary flow problems, an ideal variable
reconstruction and
flux integration scheme should produce a sum of normal components of flux
terms evaluated
at the edges of a computational cell that exactly balances bed slope source
terms evaluated at
the center of the cell, and should produce zero changes in flow momentums in
the cell. In
practice however, these two terms generally do not balance each other in most
of the low
resolution schemes, as well as in many high resolution schemes. The error
introduced by the
imbalance is often cumulative, and can significantly degrade the accuracies of
the schemes,
especially in locations where there are significant topographic variations and
controls. An
improved method of predicting fluid flow, thereby to improve production
management of
hydrocarbon resources, is desirable.
SUMMARY
[0014] A method of enhancing a geologic model of a subsurface region is
provided. The
method includes obtaining an initial bed topography of the subsurface region.
The bed
topography is defined by a plurality of cells. Each of the plurality of cells
has an elevation
associated with its cell center. The initial bed topography is represented as
a cell-centered
piecewise constant representation based on the elevations associated with the
plurality of
cells. The bed topography is reconstructed to produce a spatially continuous
surface. Flux and
gravitational force-related source terms are calculated based on the
reconstructed bed
topography. Fluxes are calculated between at least two of the cells taking
into account
variations of the bed topography across an interface between the at least two
of the cells.
Fluid flow, deposition of sediments onto the bed, and/or erosion of sediments
from the bed
are predicted using the fluxes and gravitational force-related source terms.
The predicted
fluid flow, deposition and/or erosion are inputted into the geologic model of
the subsurface
region. The geologic model is used to predict characteristics of the
subsurface region. The
predicted characteristics of the subsurface region are outputted.
[0015] The fluxes may be calculated to at least a first order of
accuracy. Reconstructing
the bed topography may comprise defining triangles by pairs of two consecutive
cell vertices
and an effective cell center. A fluid surface elevation may be reconstructed,
and flux and
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gravitational force-related source terms may be calculated based on the
reconstructed bed
topography and the reconstructed fluid surface elevation. The fluid surface
elevation may be
reconstructed using a piecewise constant scheme or a piecewise linear scheme.
Hydrocarbons
may be extracted based on the outputted characteristics of the subsurface
region. Outputting
the predicted characteristics of the subsurface region may include displaying
the predicted
characteristics. Reconstructing the bed topography may conserve mass below and
above the
spatially continuous surface. Calculating fluxes between at least two of the
cells may be
performed according to the following equation:
(0) 2
F* (1) = Fidl = 210 j *µ + ¨310(f3 Al + JAR + f5
wherein f'*(1) represents a flux per unit width between two cells at the
location that is a
distance / from a cell face middle line between the at least two of the cells,
n represents a
unit vector perpendicular to a boundary between the at least two of the cells
on an
unstructured grid, /0 represents half of the length of the boundary between
the at least two of
the cells, f*( ) represents a first expansion coefficient corresponding to a
zeroth order flux
per unit width at the center of the edge between the at least two of the
cells, /* represents a
second expansion coefficient, A L2 represents the square of a dimensionless
maximum fluid
depth change in a first of the at least two of the cells, /4* represents a
third expansion
coefficient, A R2 represents the square of a dimensionless maximum fluid depth
change in a
second of the at least two of the cells, /5* represents a fourth expansion
coefficient, A,
represents a dimensionless maximum fluid depth change in the first of the at
least two of the
cells, and AR represents a dimensionless maximum fluid depth change in the
second of the at
least two of the cells. Piecewise constant cell elevations may be updated
using the predicted
fluid flow, deposition of sediments, and/or erosion of sediments, and the
steps of obtaining,
reconstructing, calculating flux and source terms, calculating fluxes, and
predicting may be
repeated to improve accuracy of fluid flow predictions. The predicted
characteristics of the
subsurface region may include at least one of sediment grain size
distribution, porosity, and
permeability. The plurality of elevations may represent conservative
variables, which may be
updated based on the calculated source terms.
[0016] In another aspect, a computer program product having computer
executable logic
recorded on a tangible, machine-readable medium is provided, The computer
program
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product includes: code for obtaining a bed topography of a subsurface region,
the bed
topography defined by a plurality of cells, each of the plurality of cells
having an elevation
associated with its cell center; code for representing the bed topography as a
cell-centered
piecewise constant representation based on the elevations associated with the
plurality of
cells; code for reconstructing the bed topography to produce a spatially
continuous surface;
code for calculating flux and gravitational force-related source terms based
on the
reconstructed bed topography; code for calculating fluxes between at least two
of the cells,
taking into account variations of the bed topography across a face between the
at least two of
the cells; and code for predicting at least one of fluid flow, deposition of
sediments onto the
bed, and erosion of sediments from the bed using the fluxes and gravitational
force-related
source terms.
[0017] The fluxes may be calculated to at least a first order of
accuracy. Code for
reconstructing the bed topography may include code for defining triangles by
pairs of two
consecutive cell vertices and an effective cell center. Code may be provided
for
reconstructing a fluid surface elevation above the bed topography, and
calculating flux and
gravitational force-related source terms based on the reconstructed bed
topography and the
reconstructed fluid surface elevation. Code may be provided for managing
production of
hydrocarbon resources based on the prediction of the fluid flow. Code may be
provided for
outputting the predicted fluid flow, the deposition of sediments, or the
erosion of sediments.
Code may be provided for: inputting the predicted fluid flow, deposition
and/or erosion into a
geologic model of the subsurface region; predicting characteristics of the
subsurface region
using the geologic model; and outputting the predicted characteristics of the
subsurface
region.
[0018] In another aspect, a method of extracting hydrocarbons from a
subsurface region
is provided. According to the method, a bed topography of the subsurface
region is obtained.
The bed topography is defined by a plurality of cells, each of the plurality
of cells having an
elevation associated with its cell center. The bed topography is represented
as a cell-centered
piecewise constant representation based on the plurality of elevations. The
bed topography is
reconstructed to produce a spatially continuous surface. Flux and
gravitational force-related
source terms are calculated based on the reconstructed bed topography. Fluxes
between at
least two of the cells are calculated taking into account variations of the
bed topography
across a face between the at least two of the cells. At least one of fluid
flow, deposition of
sediments onto the bed, and erosion of sediments from the bed are predicted
using the fluxes
and gravitational force-related source terms. The presence of hydrocarbons in
the subsurface
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region is predicted based at least in part on the predicted fluid flow,
deposition of sediments,
or erosion of sediments. Hydrocarbons are extracted from the subsurface
region.
BRIEF DESCRIPTION OF THE DRAWINGS
[0019] Advantages will be better understood by referring to the
following detailed
description and the attached drawings, in which:
Fig. 1 is a diagram of a bed topography, a fluid surface elevation and an
unstructured grid;
Fig. 2 is a diagram of reconstructed cells in a bed topography;
Fig. 3 is a diagram of reconstructed cells in a bed topography and a
corresponding fluid
surface elevation;
Fig. 4 is a diagram that is useful in explaining calculation of flux between a
left cell and a
right cell;
Fig. 5 is a process flow diagram showing a method according to aspects
disclosed herein;
Fig. 6 is a process flow diagram showing a method according to aspects
disclosed herein;
Fig. 7 is a process flow diagram showing a method according to aspects
disclosed herein;
Fig. 8 is a simplified block diagram of a computer environment; and
Fig. 9 is a block diagram showing a tangible, machine-readable medium that
stores code to
perform methods according to aspects disclosed herein.
DETAILED DESCRIPTION
[0020] Some portions of the detailed description which follows are
presented in terms of
procedures, steps, logic blocks, processing and other symbolic representations
of operations
on data bits within a memory in a computing system or a computing device.
These
descriptions and representations are the means used by those skilled in the
data processing
arts to most effectively convey the substance of their work to others skilled
in the art. In this
detailed description, a procedure, step, logic block, process, or the like, is
conceived to be a
self-consistent sequence of steps or instructions leading to a desired result.
The steps are
those requiring physical manipulations of physical quantities. Usually,
although not
necessarily, these quantities take the form of electrical, magnetic, or
optical signals capable of
being stored, transferred, combined, compared, and otherwise manipulated. It
has proven
convenient at times, principally for reasons of common usage, to refer to
these signals as bits,
values, elements, symbols, characters, terms, numbers, or the like.
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[0021] Unless specifically stated otherwise as apparent from the
following discussions,
terms such as "obtaining", "defining", "representing", "reconstructing",
"calculating",
"producing", "taking into account", "predicting", "inputting", "using",
"modeling",
"outputting", "displaying", "conserving", "performing", "updating",
"repeating",
"extracting", or the like, may refer to the action and processes of a computer
system, or other
electronic device, that transforms data represented as physical (electronic,
magnetic, or
optical) quantities within some electrical device's storage into other data
similarly
represented as physical quantities within the storage, or in transmission or
display devices.
These and similar terms are to be associated with the appropriate physical
quantities and are
merely convenient labels applied to these quantities.
[0022] Embodiments disclosed herein also relate to an apparatus for
performing the
operations herein. This apparatus may be specially constructed for the
required purposes, or
it may comprise a general-purpose computer selectively activated or
reconfigured by a
computer program or code stored in the computer. Such a computer program or
code may be
stored or encoded in a computer readable medium or implemented over some type
of
transmission medium. A computer-readable medium includes any medium or
mechanism for
storing or transmitting information in a form readable by a machine, such as a
computer
('machine' and 'computer' are used synonymously herein). As a non-limiting
example, a
computer-readable medium may include a computer-readable storage medium (e.g.,
read only
memory ("ROM"), random access memory ("RAM"), magnetic disk storage media,
optical
storage media, flash memory devices, etc.). A transmission medium may be
twisted wire
pairs, coaxial cable, optical fiber, or some other suitable transmission
medium, for
transmitting signals such as electrical, optical, acoustical or other form of
propagated signals
(e.g., carrier waves, infrared signals, digital signals, etc.)).
[0023] Furthermore, modules, features, attributes, methodologies, and other
aspects can
be implemented as software, hardware, firmware or any combination thereof. A
component
implemented as software can be implemented as a standalone program, as part of
a larger
program, as a plurality of separate programs, as a statically or dynamically
linked library, as a
kernel loadable module, as a device driver, and/or in every and any other way
known now or
in the future to those of skill in the art of computer programming.
Additionally, the invention
is not limited to implementation in any specific operating system or
environment.
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[0024] Various terms as used herein are defined below. To the extent a
term used in a
claim is not defined below, it should be given the broadest possible
definition persons in the
pertinent art have given that term as reflected in at least one printed
publication or issued
patent.
[0025] As used herein, "and/or" placed between a first entity and a second
entity means
one of (1) the first entity, (2) the second entity, and (3) the first entity
and the second entity.
Multiple elements listed with "and/or" should be construed in the same
fashion, i.e., "one or
more" of the elements so conjoined.
[0026] As used herein, "celerity" refers to the velocity of propagation
of a wave through
a fluid or liquid, relative to the rate of movement of the fluid or liquid
through which the
wave is propagated.
[0027] As used herein, "CFL condition" refers to the Courant-Friedrichs-
Lewy condition,
which is a condition for convergence while solving certain partial
differential equations
(usually hyperbolic partial differential equations) numerically. It arises
when explicit time-
marching schemes are used for the numerical solution. As a consequence, the
timestep must
be less than a certain time in many explicit time-marching computer
simulations, otherwise
the simulation will produce wildly incorrect results.
[0028] As used herein, "displaying" includes a direct act that causes
displaying, as well
as any indirect act that facilitates displaying. Indirect acts include
providing software to an
end user, maintaining a website through which a user is enabled to affect a
display,
hyperlinking to such a website, or cooperating or partnering with an entity
who performs
such direct or indirect acts. Thus, a first party may operate alone or in
cooperation with a
third party vendor to enable the reference signal to be generated on a display
device. The
display device may include any device suitable for displaying the reference
image, such as
without limitation a cathode ray tube (CRT) monitor, a liquid crystal display
(LCD) monitor,
a plasma device, a flat panel device, an organic light-emitting diode (OLED)
device or
printer. The display device may include a device which has been calibrated
through the use
of any conventional software intended to be used in evaluating, correcting,
and/or improving
display results (e.g., a color monitor that has been adjusted using monitor
calibration
software). Rather than (or in addition to) displaying the reference image on a
display device,
a method, consistent with the invention, may include providing a reference
image to a
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subject. "Providing a reference image" may include creating or distributing
the reference
image to the subject by physical, telephonic, or electronic delivery,
providing access over a
network to the reference, or creating or distributing software to the subject
configured to run
on the subject's workstation or computer including the reference image. In one
example, the
providing of the reference image could involve enabling the subject to obtain
the reference
image in hard copy form via a printer. For example, information, software,
and/or
instructions could be transmitted (e.g., electronically or physically via a
data storage device
or hard copy) and/or otherwise made available (e.g., via a network) in order
to facilitate the
subject using a printer to print a hard copy form of reference image. In such
an example, the
printer may be a printer which has been calibrated through the use of any
conventional
software intended to be used in evaluating, correcting, and/or improving
printing results (e.g.,
a color printer that has been adjusted using color correction software).
[0029] As used herein, the "Exner equation" is a statement of
conservation of mass that
applies to sediment in a fluvial system such as a river or subsurface
hydrocarbon flow.
Specifically, it states that bed elevation increases in proportion to the
amount of sediment that
drops out of transport, and decreases in proportion to the amount of sediment
that becomes
entrained by the flow.
[0030] As used herein, "extracting hydrocarbons" refers to activities
such as determining
the general geologic structure of a subsurface region, locating potential
hydrocarbon
reservoirs, guiding the production of a previously discovered hydrocarbon
reservoir, planning
the location and timing of new wells, managing production from existing wells,
predicting
production lifetimes of wells or hydrocarbon reservoirs at various extraction
rates, and/or
other activities central or peripheral to removing hydrocarbons from a
hydrocarbon reservoir.
[0031] As used herein, "hydrocarbon" includes oil (often referred to as
petroleum),
natural gas, gas condensate, tar, and bitumen.
[0032] As used herein, "hydrocarbon reservoirs" include reservoirs
containing any
hydrocarbon.
[0033] As used herein, "machine-readable medium" refers to a medium that
participates
in directly or indirectly providing signals, instructions and/or data. A
machine-readable
medium may take forms, including, but not limited to, non-volatile media (e.g.
ROM, disk)
and volatile media (RAM). Common forms of a machine-readable medium include,
but are
not limited to, a floppy disk, a flexible disk, a hard disk, a magnetic tape,
other magnetic
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media a CD-ROM, other optical media, punch cards, paper tape, other physical
media with
patterns of holes, a RAM, a ROM, an EPROM, a FLASH-EPROM, or other types of
memory
chips or cards, a memory stick, and other media from which a computer, a
processor or other
electronic device can read.
[0034] As used herein, "MUSCL" refers to Monotone Upstream-centered Schemes
for
Conservation Laws, which are finite volume methods s that can provide highly
accurate
numerical solutions for a given system, even in cases where the solutions
exhibit shocks,
discontinuities, or large gradients.
[0035] In the oil and gas industry, solving water flow and sediment
transport equations is
useful in constructing physics process-based models for simulations of
formation and
evolutions of sedimentary basins and oil and gas reservoirs, and the
subsequent building of
the geologic models. Such models may be useful in predicting fluid flow
parameters such as
erosion or deposition, and in other activities related to hydrocarbon
extraction. In geologic
modeling, a digital representation of the detailed internal geometry and rock
properties of a
subsurface earth volume, such as a petroleum reservoir or a sediment-filled
basin is built. In
the oil and gas industry, geologic models provide geologic input to reservoir
performance
simulations which are used to select locations for new wells, estimate
hydrocarbon reserves,
plan reservoir-development strategies, and/or perform other hydrocarbon
extraction activities.
The spatial distribution of permeability may assist in characterizing
reservoir performance,
and, together with other rock and fluid properties, may determines the
producibility of the
reservoir. For sandstone reservoirs, the spatial distribution of permeability
is a function of
the grain-size distribution of sands which compose the reservoir, the
compartmentalization of
those sands by barriers of finer grained material, and the mineralogy and
burial history of the
reservoir.
[0036] Most geologic models built for petroleum applications are in the
form of a three-
dimensional array of model blocks (cells), to which geologic and/or
geophysical properties
such as lithology, porosity, acoustic impedance, permeability, and water
saturation are
assigned (such properties are referred to collectively herein as "rock
properties"). The entire
set of model blocks represents the subsurface earth volume of interest. The
goal of the
geologic-modeling process is to assign rock properties to each model block in
the geologic
model.
[0037] In one aspect, a bed topography reconstruction scheme and a flux
integration
scheme balances flux gradients and corresponding bed slope source terms to
second orders of
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accuracy over arbitrary bed topographies. The combination of the bed
topography
reconstruction scheme and the flux integration scheme leads to more accurate
results when
compared to known methods, with only a minor increase in computational effort.
The
method may be used with cell-centered discretization schemes where both flow
variables as
well as bed topographic information are represented in the center of cells.
[0038] Relative to most of the known discretization schemes in which the
bed topography
is represented in cell vertices, a cell-centered discretization scheme in one
aspect may be the
simplest one that attempts to ensure the conservation of mass for both solids
below the top
bed surface and the liquid above. Such a constraint may provide improved
results when the
flow and transport equations are coupled with sediment erosion, deposition and
the sediment
mass conservation equations on the bed, such as the Exner equation, to solve
bed evolutions
in both shallow and deep water environments. In a cell-centered discretization
scheme, the
amount of erosion, deposition and cell elevation change are all calculated
using the flow and
sediment information associated with the center of the cell, without any need
of extrapolation
among neighboring cells.
[0039] For known discretization schemes in which bed topography is
represented in cell
vertices, some extrapolation and averaging of either flow variables, or
sediment erosion or
deposition rate are typically useful in the calculation of elevation changes
on cell vertices,
which could cause numerical diffusion for the bed topography. Therefore, a
cell-centered
discretization scheme in accordance with the present techniques provides an
advantage over
known schemes that employ data along cell vertices, in that the numerical
diffusion of cell
elevations due to the averaging and extrapolation among neighboring cells is
reduced or
eliminated when the equations related to sediment transport, erosion and
deposition processes
are added to flow equations.
[0040] Fig. 1 is a diagram of a bed topography, a fluid surface elevation
and an
unstructured grid that is useful in explaining the operation of an embodiment.
The diagram,
which is generally referred to by reference number 100, includes a bed
topography 102, a
fluid surface elevation 104 and an unstructured grid 106. The unstructured
grid 106 includes
a plurality of two-dimensional cells. A first cell 108 is a polygon that is
bounded by line
segments ABCDE. A second cell 110 is a polygon that is bounded by line
segments AGHB.
A third cell 112 is a polygon that is bounded by line segments BHIJC. A fourth
cell 114
comprises a polygon that is bounded by line segments CKLD. A fifth cell 116 is
a polygon
that is bounded by line segments DMNOE. A sixth cell 118 is a polygon that is
bounded by
line segments AEF.
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[0041] An embodiment relates to a cell-based discretization scheme in
which values of
variables are associated with cell centers rather than at their vertices.
Spatial variability of
each variable within each cell is not stored and may only be reconstructed
using different
reconstruction schemes. The simplest of such schemes is the piecewise constant
reconstruction scheme. In this scheme, the value for each variable remains
constant within
each cell and equals the value that is stored in the cell. In Fig. 1, the
results of the piecewise
constant reconstruction of the bed topography 102 and the fluid surface
elevation 104 are
shown together with the unstructured grid 106. One prominent feature to note
is the
discontinuities of the reconstructed values across cell edges. For example,
the elevations
across the cell edge BC on the underlying grid 106, which correspond to the
edges marked
B1C1 and B3C3 in the bed topography 102, for the first cell 108 and the third
cell 112
respectively, are discontinuous. Moreover, the edges B1C1 and B3C3 are located
at different
positions in the z-direction relative to the underlying grid 106. Similarly,
the fluid surface
elevation 104 shows that the corresponding areas labeled as lines B1*C1* and
B3*C3* are
also discontinuous in the z-direction.
[0042] An embodiment produces continuous bed surfaces, eliminates the
discontinuities
as described above, and in the process, conserves the mass in volumes bounded
by
corresponding areas of the bed topography 102 and the fluid surface elevation
104, as well as
volumes bounded by corresponding areas of the bed topography 102 and the
unstructured
grid 106. In other words, for each cell, the amount of mass that is enclosed
between any
arbitrary surface below or above the bed topography 102 remains the same
before and after
the reconstruction of the bed surface in that cell.
[0043] Fig. 2 is a diagram of reconstructed cells in a bed topography.
The diagram, which
is generally referred to by reference number 200, includes a bed topography
202 and an
unstructured grid 206. The unstructured grid 206 is a plurality of cells. A
first cell 208 is a
polygon that is bounded by line segments ABCDE. A second cell 210 is a polygon
that is
bounded by line segments AGHB. A third cell 212 is a polygon that is bounded
by line
segments BHIJC. A fourth cell 214 is a polygon that is bounded by line
segments CKLD. A
fifth cell 216 is a polygon that is bounded by line segments DMNOE. A sixth
cell 218 is a
polygon that is bounded by line segments AEF.
[0044] The bed topography 202 includes two cells that have been
reconstructed. In
particular, the bed topography 202 shows a reconstructed first cell 208',
which corresponds to
the first cell 208 of the unstructured grid 206, and a reconstructed third
cell 212', which
corresponds to the third cell 212 of the unstructured grid 206.
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[0045] The surfaces defined by triangles xleAmBm, xie BmCm, xieCmDm,
xieDmEm, and
XieEmAm are the reconstructed surfaces for the first reconstructed cell 208'.
The surfaces
defined by triangles x3eCmBm, x3eBmHm, x3eHmIm, x3eImJm and x3eJmCm are the
reconstructed surfaces for the third reconstructed cell 212'. In Fig. 2, the
points Am, Bm,
Cm, Dm, Em, Hm, Im, and Jm represent the effective elevations at the vertices
A, B, C, D, E,
H, I, and J respectively. The points xie and X,3e are the effective cell
center elevations for the
first reconstructed cell 208' and the third reconstructed cell 212'
respectively.
[0046] The points xib and xi represent the projections of the point xie
on the original
piecewise constant cell bed surface A1B1C1D 1E1 and the gridding surface
ABCDEF. The
points x3b and x3 are the projections of the point x3e on the original
piecewise constant cell
bed surface C3B3H3I3J3 and the gridding surface CBHIJ.
[0047] As set forth above the volume of regions between the bed
topography 202 and the
unstructured grid 206 is conserved. For example, as shown in Fig. 2, the
volume enclosed by
the surface ABCDE and piecewise constant bed surface A1B1C1D1E1 are the same
as the
volume enclosed by the surface ABCDE and the reconstructed bed surface, which
includes
five triangles xleAmBm, xie BmCm, xieCmDm, xieDmEm, and xieEmAm. Similarly,
the
volume of the fluid above the bed surface in the cell is also conserved before
and after the
reconstruction of the bed.
[0048] Reconstruction of a bed topography includes three steps:
calculating the cell
center, calculating the effective elevation at each cell vertex and
calculating the effective
elevation for the cell center. Each of these steps is described in detail
below.
[0049] The first step of reconstructing a bed topography is to calculate
the cell center.
For any cell, if there are n vertices with xy-coordinates xi and y for i =
1,2,3,...n , the xy-
coordinate of the cell center xe and y may be given by equations (1) and (2):
1 "
x=--x i (1)
n
1 "
Ye = E Y, (2)
n
[0050] The second step of reconstructing a bed topography is to
calculate the effective
elevation for the cell center. Each of the vertices of a given cell is shared
among a number of
cells. If a vertex is shared by m cells, and each cell j, where j = 1,2,3,...m
, has an averaged
elevation of z1, the effective elevation at each of the cell vertices may be
calculated using the
following equation (3):
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Z .
Tc11
j=1 j
Z 1. = (3)
v,n1 1
44¨d
J=1
where
d = V(xi ¨ xe,1)2 + (yi ¨
is the distance from a vertex to the center of the j th cell that
shares the vertex.
[0051]
The third step of reconstructing a bed topography is to calculate the
effective
elevation for the cell center. As is shown in Appendix A hereto, the effective
cell center
elevation may be given by equation (4):
(i)
¨ aI ____ x(i)A (i) __ b(i) __ y(i) Aci) Aci)
c
i)
( c
C c(i)
z = ________________________________________________________ (4)
(i) b(i) d(i)
(i) c
______________________________________________
C C (i)
[0052] In equation (4), A(i) is the area projected to the xy plane by the i
th triangle Qi in
the cell defined by any of the two consecutive vertices i, i +1 (or vertex 1
if i = n), and the
cell center. An example is triangle AmBmxie. Its projected area is the area of
the triangle
ABxi. In the same equation, Xe(i) and ye(i) are the x and y coordinates of the
center of that
triangle, respectively, and the values of 4) , a, b, b, c(i) , di(i) , 4) may
be given by
equation (A6) and equations (A8-A13) in Appendix A.
[0053]
Once z, is found, the elevation of any point in a cell with coordinates (x, y)
on the
reconstructed bed surface may be given by equation (5):
¨x + b(i) y + d(i)
z = ________________________________________________________ (5)
c (i)
with (x, y) e Q. . In above equation, a(i) , b(i) , c(i) , and d (i) are given
by equations (A4-A7),
respectively, in Appendix A.
[0054]
The reconstructed bed surface elevation z can be obtained in another way. For
each triangle Qi , its surface gradient (S,S)= õ
¨,¨ may be given by equations (6)
x ,y aX ay
and (7):
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(1) = ye)zi+i (Ye +1 )z (Yi+1 (6)
'qx
xeYi+i ¨ Yi+1 xeY, xi Ye ¨
(i) = (x, ¨ xe)z, + (x, ¨ x1 )z + (xi, ¨ xi)z e
(7)
'17 y
Xc YL-F1 ¨ XL YI xeY, xi Ye ¨ Ye
[0055] Therefore,
(8)
It can be shown that equations (5) and (8) are equivalent.
[0056] Fig. 3 is a diagram of reconstructed cells in a bed topography
and a corresponding
fluid surface elevation. The diagram, which is referred to generally by
reference number 300,
includes a bed topography 302 and a fluid surface elevation 304. The bed
topography 302
comprises the reconstructed cells described above with reference to Fig. 2.
[0057] In the diagram 300, the surfaces A1*B1*C1*D1*E1*, A2*G2*H2*B2*,
B3*H3*I3*J3*C3*, D4*C4*K4*L4*, 05*E5*D5*M5*N5*, and F6*A6*E6* of the fluid
surface elevation 304 are the piecewise constant representation of a fluid
surface. The
surface Ae*Be*Ce*De*Ee* is the piecewise linear representation of the fluid
surface. The
fluid surface may be represented by a number of different schemes. The two
simplest
schemes are the piecewise constant scheme and the piecewise linear scheme.
[0058] A piecewise constant scheme may assume that the fluid surface
elevation
y) is constant everywhere in a cell. Such a piecewise linear scheme may take
many
different forms. For example, one form is set forth in the equations (9),
(10), (11), and (12):
j(x, y) = + S(x ¨ )0+ Sj,(y ¨ ye) (9)
Since
(x ¨ xe)dxdy = f xdxdy ¨ xe f dxdy =0 (10)
and
(y ¨ ye)dxdy = f ydxdy ¨ ye f dxdy = 0 (11)
it may be shown that
j(x, y)dxdy = dxdy (12)
for any values of S and S. . Accordingly, mass conservation is strictly
preserved in this
scheme.
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[0059] Another aspect is that calculations of the inter-cell fluxes
correspond to the
reconstruction scheme described above. The general form of the governing
equations for
both shallow and deep water turbidity current can be written as equation (13):
¨k f U-dxdy +f, (Pdy - adx) = f gg dxdy + f g fdxdy (13)
at acl fl
fl
where U is the vector of conservative variables, P and a are the vectors that
are related to
the conservative form of the fluxes in x and y directions respectively, gg is
the vector of the
source terms related to the gravitational forces, and gf is the vector of the
source terms
originated from other than gravitational forces. Only the first three terms
may need to be
calculated using inventive methods disclosed herein. The calculation for the
last term may
use existing methods and can be readily determined.
[0060] The specific form of U, F, a and gg may vary from different
formulations for
shallow and deep water flows. One embodiment may, however, be applied with
respect to all
of the forms.
[0061] Apart from the source term ,c'f, , the general form for the
governing equations for
both shallow water flows and the turbidity currents in deep water environment
can be written
in a unified fashion as shown below in equations (14)
(15), (16) and (17), where
1 uxh
u h
Y
h
U= (14)
hCk
hK
i
i
Ux2h +1 (R,, + RC)gh 2
2
uxUyh
p_. hux (15)
hu xC k
hu xK
\ /
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i
u xuyh
u 2h + ¨1(R + RC)gh2
Y 2 w
o = huY (16)
huyC k
hu K
Y /
and
i az
¨(Rw + RC)gh ¨
ax
¨ (Rw + RC)gh ¨az
ay
AS' , = 0 (17)
0
0
\ 1
[0062] In the above equations, ux and uy are the x - and y - components of
the depth
averaged flow velocity respectively, h is the depth of the turbid flow, z is
the bottom
elevation, C k is the volumetric concentration of the sediments in the k th
grain size bin and
equation (18)
n
C =EC k (18)
k=1
represents the total sediment concentration. In equations (15) and (16), g
represents the
gravitational constant, and R is the submerged specific weight for the
sediments. R may be
represented by
R ¨ Ps ¨ Pw (19)
Pw
In equation (19), ps and pw are the sediment and water density respectively.
In equations
(14)-(16), K is the turbulent kinetic energy. The definition of Rw in
equations (15) and (16)
may be given by
R = Pw ¨ P a (20)
w
Pw
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where pa is the density of the ambient fluid. In a deep water environment, the
ambient fluid
is the same as the flowing fluid, therefore Rw can be represented by
Pw¨ Pa Pw¨ Pw
R = = =0 (21)
w
Pw Pw
On the other hand, in a shallow water environment, the ambient fluid is air,
which has a
density much smaller than the water density, where Pa <<i'. Therefore Rw can
be
represented by
R =Pw¨ Pa =1--Pa 1 (22)
w
Pw Pw
for the shallow water environment.
[0063] Without the loss of generality, the flow properties ux , u,, Ck ,
for k = 1525...n 5 and
K are all treated as piecewise constant in the cell in the descriptions of the
method given
below. However, the method described below can be easily applied without
treating the flow
properties ux 5 uy, Ck , for k = 1,2,...n, and K as piecewise constant in the
cell. In those
situations, any well-known extrapolation schemes for ux, uy, Ck , for k =
1525...n 5 and K can
be used with the disclosed aspects.
[0064] In one aspect, fluxes may be calculated across cell faces. An
example of a cell
face is the surface BmCmCe*Be* shown in Fig. 3. The calculation of flux across
a cell face
is described in detail below with reference to Fig. 4.
[0065] Fig. 4 is a diagram that is useful in explaining calculation of
flux between a left
cell and a right cell. The diagram is referred to generally by reference
number 400. The
diagram 400 comprises bed topography 402 and a fluid surface elevation 404.
The bed
topography 402 represents a subset of the bed topography 302 shown in Fig. 3.
In particular,
the representation of the bed topography 402 corresponds to the reconstructed
cells of the bed
topography 302. The fluid surface elevation 404 represents a subset of the
fluid surface
elevation 304 shown in Fig. 3. Without losing generality, a first cell 406 of
the fluid surface
elevation is denoted as a left cell. A second cell 408 of the fluid surface
elevation 404 is
denoted as a right side cell. Calculating the fluxes of fluid flowing between
the left cell and
the right cell is described in detail below.
[0066] The reconstructed bed surfaces of the topography bed 402, which
may be obtained
using a reconstruction scheme disclosed herein, are continuous across the cell
faces. The
fluid surfaces of the fluid surface elevation 404 are, however, often
discontinuous, as shown
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in Fig. 4. Therefore the water depths at the cell face in two cells are often
different from each
other. Also, the degree of the discontinuities may be different at the
different locations on the
cell face. The expression hi, = hL (/) may be used to denote the water depth
at the cell face in
the left cell, and at the location / distance away from the center line 410 of
cell face as shown
in Fig. 4. Similarly, hR = hR (1) denotes the corresponding water depth in the
right cell. It
can be shown that
A /
h_L(1)=4,( ) + AL ¨/ = h( ) (1+ ¨) = 1,4 ) (1+ ALi) (23)
L h"
0 L 0
A /
hR(l) = hR( ) + AR == hR(0)(1 + ¨) = hR( )(1+ ARi) (24)
R 0
where
k ) = ¨1 ¨ z, +
L ,i+1 Z L ,i+1) (25)
2 "
h( ) = ¨1(R ,z j ¨zR i+ ,i+1 ZR ,i+1) (26)
R 2 ,
A (J1, ,i+1 Z L ,i+1) L Z ,i)
(27)
L
2
A = (J = ¨ z = ) (JR,i zR,i)
(28)
2
[0067] In the above equations, /0 is the half cell edge length for cell
i or i +1,
/
= ¨ (29)
is the dimensionless distance from the cell face middle line,
A
A = L (30)
L h(0)
"L
is the dimensionless maximum fluid depth change in the left cell at the cell
face, and
A
AR = R (31)
(o)
R
is the dimensionless maximum fluid depth change in the right cell at the cell
face.
[0068] The fluxes across the cell face can be evaluated using many
different solvers.
Here a solver known an HLL solver ('HLL' being an acronym for Harten, Lax, and
van Leer)
is used as an example. The HLL solver obtains an approximate solution for the
inter-cell
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numerical flux directly by assuming a particular wave configuration for the
solution and
calculating the wave speed (Harten, A., Lax, P., and van Leer, A., "On
upstream differencing
and Godunov-type scheme for hyperbolic conservation laws", Soc. Indus. Appl.
Math. Rev.,
25(1), 35-61, 1987.). The HLL solver has proven stable even when running long
enough to
generate highly complicated stratigraphy.
[0069] Assuming
FGL = (PL, OL) , (32)
the flux across the cell face may be given by equation
s, FGL = ii ¨ sL FGR = /7/ + sLsR (i/R ¨ I i L)
(33)
SR ¨ SL
where the celerities of the waves at the cell face are 5L and5R , P* (1)
represents the flux per
unit width between two cells at the location that is a distance / from a cell
face middle line
between the at least two of the cells, and ii represents the unit vector that
is perpendicular to
a boundary between the at least two of the cells on an unstructured grid. P*
(1) is the same as
P* with its variable / shown.
[0070] Other variables helpful in understanding disclosed aspects include
the following:
/0, representing half of the length of the boundary between the at least two
of the cells; 7(0),
representing a first expansion coefficient corresponding to a zeroth order
flux per unit width
at the center of the edge between the at least two of the cells, the first
expansion coefficient
being computed according to equation (B48) in Appendix B; j'3*, representing a
second
expansion coefficient that is computed according to equation B51 in Appendix
B; AL2 ,
representing the square of a dimensionless maximum fluid depth change in a
first of the at
least two of the cells; j:4*, representing a third expansion coefficient that
is computed
according to equation (B52) in Appendix B; AR2 , representing the square of a
dimensionless
maximum fluid depth change in a second of the at least two of the cells; /5*,
representing a
fourth expansion coefficient that is computed according to equation (B53) in
Appendix B;
AL, representing a dimensionless maximum fluid depth change in the first of
the at least two
of the cells; and AR , representing a dimensionless maximum fluid depth change
in the second
of the at least two of the cells.
[0071] Separating constant states in the solution of the local Riemann
problem can be
calculated using Toro's two expansion approach as set forth in E. F. Toro,
"Riemann
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Problems and the WAF Method for Solving Two-Dimensional Shallow Water
Equations",
Phil. Trans. Roy. Soc. London, A338: 43-68, (1992), and E. F. Toro, "The
Weighted Average
Flux Method Applied to the Time-Dependent Euler Equations", Phil. Trans. Roy.
Soc.
London, A341: 499-530 (1992). Toro's two expansion approach states that for
wet cells,
SL = min(-4, = ii - + RC)gh,,u* - V(R,v+ RC)gh*) (34)
SR = min(qR = ii + V(R + RC)ghR,u* + Al(Riv+ RC)gh*) (35)
where
-4,=(ux,,u,,,) (36)
and
= (uxR t yR) = (37)
[0072] In the above equations, ux, and uy, are the x and y components of
the flow
velocity in the left cell, respectively. Similarly, uxR and uyR are
respectively the x and y
components of the flow velocity in the right cell.
[0073] In equations (34) and (35),
u. = (4, + R) = + 11(Rõ+ RC)gh, - Al(Rõ+ RC)ghR (38)
2
and
V(Rõ + RC)gh* =-1(11(Rõ+ RC)gh, +11(Rõ+ RC)ghR)+ ¨1(4, -4R)= (39)
2 4
[0074] Because the fluid depths in both left and right cells vary with the
distance / from
the center of the cell face, and the differences of the fluid depths across
the cell face also
varies with 1, the value of flux P* = ñ in equation (33) varies with 1. The
calculation of the
-
total flux across the cell face requires the evaluation the integral F*
(1)=fidl . To avoid
computing the integral numerically, which may add significant amount of
computational
overhead to the scheme, equation (33) is first Taylor expanded to the order of
o(A2, ) , o(A)
or o(ALAR ) and then the result is integrated over i analytically. Detailed
derivation is given
in Appendix B. The final result is
jF (1)= =210:i*" +-10(f3*A,,+ r4 AR f5*ALAR (40)
3
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where the expressions for evaluating *( ) , j; and j'5* are given in
equations (B48) to
(B53) in Appendix B.
[0075] The current reconstruction scheme subdivides a cell into a number
of triangles.
Therefore the integration of the source term in equation (13) is also
subdivided into the
corresponding number of parts before each parts are summed together to yield
the final
results. The detailed steps of the calculation are shown in Appendix C. The
result is
Lggdxdy
¨
S(1) (Rw + RC)[( ¨ zc)A ¨ Sq(x1))(xe(1) ¨ xc)A (S4tõ ¨ S)(y
c(1) ¨ yc)A
(E
E¨ S,v(1)(Rw + RC)[( ¨ ze)Aõ, ¨ Sq(x1))(xe(1) ¨ xe)Aõ,+ (S4tõ ¨
S,v(1))(ye(1) ¨ ye)21]
0
0
0
0
(41)
[0076] Fig. 5 is a process flow diagram showing a method 500 according
to disclosed
techniques and methodologies. At block 502 an initial bed topography of a
subsurface region
is obtained. The bed topography is defined by a plurality of cells. Each of
the plurality of
cells has an elevation associated with its cell center. This elevation may be
a fluid height
elevation, bed elevation, or both. Other conservative variables, such as
sediment volumes,
may be stored in the center of one or more cells. The values may be stored in
a memory (as
discussed further herein) associated with each cell center. As shown in the
figures, a two-
dimensional unstructured grid of cells, with elevations associated with the
cell centers, may
form and be represented as a piecewise constant bed topography (and/or a
piecewise constant
fluid surface topography) that may be expressed or displayed as a three-
dimensional model of
cells showing bed and/or fluid elevations.
[0077] Using the elevations stored at block 502, at block 504 the piecewise
constant bed
topography is reconstructed to produce a spatially continuous surface for flux
and source
term calculations. At block 506, a piecewise constant fluid topography may be
reconstructed
based on the spatially continuous surface. At block 508 flux between cells is
calculated using
the reconstructed bed topography. As an example, the flux calculation may be
performed as
described above with reference to Fig. 4 and may include calculating flux
terms based on the
reconstructed bed topography. For example, the flux calculation may take into
account
variations of the bed topography across a face between the cells. At block
510, gravitational
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force-related source constants may be calculated based on the reconstructed
bed topography.
The reconstructed fluid topography may additionally be used in calculating the
gravitational
force-related source constants. At block 512, other source terms for the cells
may be
calculated, if needed.
[0078] The conservative variables stored at block 502 may be updated and
primary
variables computed, as shown at block 514. Based on the updated variables,
fluid flow
parameters such as erosion and deposition may be predicted, as shown at block
516. The
piecewise constant cell elevations may additionally be updated. As shown in
Fig. 5, the
method may return repeatedly to block 502 to improve accuracy of fluid flow
predictions.
[0079] Figure 6 shows a method 600 that uses predictions of fluid flow,
erosion and/or
deposition as an input to a geologic model. At block 602 the method receives
such
predictions as are made at block 516 of Figure 5. At block 604 the predictions
are input into a
geologic model. The geologic model may be one as is described previously
herein. At block
606 the geologic model is used to predict geologic characteristics (such as
sediment grain size
distribution, porosity, permeability, connectivity, and the like) of a
subsurface region. At
block 608 values for the predicted characteristics are outputted.
[0080] Figure 7 shows a method 700 that uses predictions of fluid flow,
erosion and/or
deposition, and/or geologic characteristics to extract hydrocarbons from a
subsurface region,
which may include a hydrocarbon reservoir or field. At block 702 the method
receives the
predictions made at block 516 and/or block 608. At block 704 the presence of
hydrocarbons
in a subsurface region is predicted based at least in part on the predictions
made at block 516.
At block 706 hydrocarbons are extracted from the subsurface region. The
activities defined
by the term "extracting hydrocarbons" may be broadly interpreted as set forth
herein.
[0081] Figure 8 depicts a block diagram of a computing environment 800
that may
implement one or more of the disclosed methods. Computing environment 800
includes a
system computer 802, which may be implemented as any conventional personal
computer or
workstation, such as a UNIX-based workstation or any other computing clusters
or
supercomputers. The system computer 802 is in communication with disk storage
devices
804, 806, and 808, each of which may be any known type of computer-readable
storage
media such as external hard disk storage devices that are either directly
connected to the
system computer or accessed using a local area network or by remote access.
Although disk
storage devices 804, 806, and 808 are illustrated as separate devices, a
single disk storage
device may be used to store any and all of the program instructions,
measurement data, and
results as desired.
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[0082] In one aspect, the input data are stored in disk storage device
806. The system
computer 802 may retrieve the appropriate data from the disk storage device
806 to perform
program instructions corresponding to the methods described herein. The
program
instructions may be written in a computer programming language, such as C++,
C#, Java and
the like. The program instructions may be stored in a computer-readable
memory, such as
program disk storage device 808. System computer 802 presents output primarily
onto a
text/graphics display 810, or alternatively to a printer 812. The system
computer 802 may
store the predictions or other results of the methods described herein on disk
storage 804, for
later use and further analysis. A keyboard 814 and a pointing device (e.g., a
mouse, trackball,
or the like) 816 may be provided with the system computer 802 to enable
interactive
operation. The system computer 802 may be located at a data center remote from
the
reservoir. Additionally, while the description above is in the context of
computer-executable
instructions that may run on one or more computers, those skilled in the art
will recognize
that the subject matter as claimed also can be implemented in combination with
other
program modules and/or as a combination of hardware and software.
[0083] Fig. 9 is a block diagram showing a tangible, machine-readable
medium that
stores code adapted to perform method disclosed herein. The tangible, machine-
readable
medium is generally referred to by the reference number 900. The tangible,
machine-
readable medium 900 may comprise one or more hard disk drives, a DVD, a CD or
the like,
and may correspond to disk storage device 806 in Figure 8. The tangible,
machine-readable
medium 900 stores computer-readable code 902 for obtaining a bed topography of
a
subsurface region, where the bed topography is defined by a plurality of
cells. Each of the
plurality of cells has an elevation associated with its cell center, and the
bed topography and
elevations may be stored in a memory location in disk storage devices 804,
806, and/or 808.
Code 904 represents the bed topography as a cell-centered piecewise constant
representation
based on the cell center elevations. Code 906 reconstructs the bed topology as
a spatially
continuous surface. Code 908 calculates flux and gravitational force-related
source terms
based on the reconstructed bed topography. The tangible, machine-readable
medium 900 also
stores code 910, which calculates fluxes between at least two of the cells
taking into account
variations of the bed topography across a face between the at least two of the
cells. Code 912
predicts fluid flow, deposition of sediments onto the bed, and/or erosion of
sediments from
the bed using the fluxes and the gravitational force-related source terms.
Furthermore, the
tangible, machine-readable medium 900 may contain or store code relating to
one or more of:
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CA 02754736 2015-11-16
inputting predictions into a geologic model of the subsurface region, using
the geologic
model to predict characteristics of the subsurface region, outputting the
predicted
characteristics of the subsurface region, predicting the presence of
hydrocarbons in the
subsurface region, extracting hydrocarbons from the subsurface region, and any
other actions
or steps disclosed herein.
[0084] The invention may have broad applications in the oil and gas industry,
as well as
many other areas such as environmental protection and management, flood and
hurricane
control, and flood and hurricane forecasting.
[0085] The disclosed embodiments and methodologies may be susceptible to
various
modifications and alternative forms and have been shown only by way of
example. For
example, the formulas and expressions disclosed herein, and the variables
contained in those
formulas and expressions, may be modified or expressed in alternative forms.
The disclosed
embodiments and methodologies are not intended to be limited to the particular
embodiments
disclosed herein, but include all alternatives, modifications, and
equivalents. The scope of the
claims should not be limited by particular embodiments set forth herein, but
should be construed
in a manner consistent with the specification as a whole.
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APPENDIX A
[0086] Cell center effective elevation calculation.
[0087] Consider a cell with n vertices arranged counter clockwise from i
= 1,2,3,... to n.
Each vertex i has a coordinate of (xi, yi ) . The coordinate of the center of
the cell (xõ ye) is
then given by equation (1) and (2). The triangle defined by any two
consecutive vertices i,
i +1 (or vertex 1 if i = n), and the cell center is refered to as Qi .
[0088] Denote the effective elevation of each vertex i as zi and of the
cell center as z.
Use to denote the piecewise constant representation of the cell
elevation. The problem is to
find the value for z, such that the cell averaged elevation over the surfaces
defined by Qi for
i = 1,2,3,...n equals . In other words, solve z c so that
I zdxdy = YE , (Al)
where z = z(x, y) is the reconstructed cell elevation at location (x, y) .
[0089] In triangle Q., z should satisfy the following equation
x y z 1
xe ye z, 1
=0 (A2)
xi yi zi 1
xi+1 Yi+1 zi+1 1
In equation (A2) and throughout Appendix A, we define i +1 = 1, if i = n.
[0090] From above equation, it may be shown that
z = ______________________________________________________ (A3)
c(i)
where
4) = 4) + 4)z, (A4)
b(i) = bi(i) + bz (A5)
(i) = xeYi xeYi+i xi.Ye xi Yi-Fi xi-Fi Ye ¨ x1y (A6)
= 4) + dz, (A7)
and
cif = yezi ¨ yezi i + yz1 ¨ (A8)
(i)
-
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NI) = Xezi ¨ xezi i ¨ x1z (A10)
= xi ¨ xi+1 (A11)
= xe ¨ Ye (Al2)
= (A13)
[0091] Substitute equation (A3) for equation (Al), using the following
properties where
f xdxdy = XciA, (A14)
and
f ydxdy = YciA, (A15)
the effective cell center elevation is given by
(i) (i) (i)
fE Au) ¨ E ______________ x
a(i) c
(i)A ______________________________ y
ci) (i)A(i) __ A(i)
c(i) c
c c(i)
Ze= _________________________ b(i) (A16)
(i)
(i) d(i)
r az x(i)Aci) 2 y(()A(i) 2 A(i)
c
C(i)
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APPENDIX B
[0092] Calculation of HLL fluxes.
[0093] To enable an analytical integration of the equation (33) with
regard to the variable
1, the HLL flux function shown in the right side of equation (33) is first
Taylor expanded to
the second order of o(/X), o(A) or o(ALAR ) .
[0094] Substituting equations (23) and (24) into equations (14) to (17)
and expanding the
results to the second order, we obtain
FGL = FGL + fiAL/ + f2A2L/2 (B1)
FGR = FGR + f3AR/ + f4A2R/2 (B2)
Oz. =U'L kiAL1 (B3)
CI' R=UR k2AR1 (B4)
where
¨FGL(0) = (F7 ), G-L( ))
1 h (0)
x ,L
1 2h1 ) + ¨ (Rw + RC L)ghl())2 Ux,LU y,L"L
2
+ ¨1(R + RC L)gh(L )2
Ux,LU y,Lh(L(1) y,L L 2 w
h( )u h( )u (B5)
L x ,L 5 L y,L
hr)U x ,LC k hi )uy,LCk,L
L x,L L L y,L L
1 2 (0)(0)
Ux,L "L (Rw RC L)ghr x,LU y,L"14L
x ,LU y ,Lhr) y ,L2 h(L(1) (Rw + RC L)ghr)2
u x,Lhr) u h( )
fi - 5 y,L L (B6)
(0) U C h"
Ux,LCk,LHL y,L k,L L
cl'4 )K
y,L L L
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PCT/US2010/021641
1 1 0
2
0
¨ (R,, + RC jgh0)2,
2
.72 = 0 0
(B7)
,
0 0
0 0
\ 1
F G(Ro) = (Pf, ) , O0)
i 1 u x,Ru j,,Rh(;)
u ,R2 hP + ¨ (R + RC R)g14)2
2 w 1
ux,Ruj,,c) uYR 2c) +-(Ri, + RC R)ghP2
' 2
(B8)
hP u j, ,R
,
k(I)Ux,RCk,R k())21j,,RCk,R
\ 1
lux,R2h(R ) +(R +RCR)g14)2 ux,Ruj,,Rc)
ux,Ruj,,R14) u j, , R 2 14) + (R + RC R)ghP2
u x,RhP u h( )
y RR
13 =(B9)
U x ,RC k ,R14 , ) U C h( )
k,R R
\
U x ,RK RC) UyR KR h"R , 1
1 1\
¨ (R + RC R)ghP2 01
2 w
0 ¨ (R + RC R)ghP2
2 w
.74 ¨ 0 0
(B10)
,
0 0
0 0
\ 1
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L
u h(0)
y,L L
V)
_ h
L (B11)
Li L ¨ (0) i, C k ,L
\ L L)
1h(0)
U x ,L L
u h"
y,L L
h"
_ g
_
(B12) i ¨ C h( )
k ,L L
K h"
\ LL /
1h( )
U x ,R R
u h"
y,R R
h( )
= R (B13)
'-) R
h12 ) C k ,R
hm K
\ R R)
1h( )
U x ,R R
u h"
y,R R
h( )
g'2. =(B14)
C h( ) =
k ,R R
K hm
\ RR /
[0095] Write sL and SR as
SL = s( ) +cslALI Cs2ARI C3 A2 1.2 c4 A2 12 + cs9ALAR1 (B15)
L s L s R
SR = s(R ) cs5AL1+ cs6AR1+ cs7A2Li2 cs8A2Ri2
cs1oALAR12 . (B16)
The expressions for the expansion coefficients in above equations are shown
below.
[0096] Substituting equations (23) and (24) in equations (38) and (39)
results in
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. 1
u = -2(4L -4R ) = ji
lA i 11A n2 1 _____________ 1 A i 1 IA P2-
(13
+ Al(R,, + RC)g14 ) 1+ ¨' ¨ ¨ ¨(R + RC)gh, ) 1+ ¨' ¨ ¨
2 14 ) 8 1,
4) j 2 h? ) 8 hP j
17)
,.. i ,..2.
1 A' AL'
V(R,, + RC)gh* = ¨1 V(R,, + RC)gh(, ) 1+--L 1
2 2 1,4 ) 8 1,40) j
1 i ________________________________ 1+ 1 AR/
4,
2 w 2 h( ) 8
R R y
(B18)
[0097]
The next step is to use the results in the above equations and compare values
of
the following expressions
-4,, = n - V(R,, + RC)ghL
VAI(Rw + RC)gh(L ) ,, 1 (R + RC)g14 ) ,,
= 4,, = h - Al(R,, + RC)g14 ) AL' + ______________ A2L/ 2
2h(L ) 8 1,4 )2
(B19)
and
u* ¨ V(Rõ + RC)gh* = (-41 4 +-43 -4R)= Ti + -21 V(Rõ+ Rc)gh(, ) - -23 V(Rõ +
RC)gh)
1 11(R + RC)gh(L ) 3 V(R + RC)gh(R )
+¨AL/ ______________________________________________________ AR/
4 1,4 ) 4 hP
I Al(R + RC)g14 ) A2 i2 3 V (R + RC)ghP A2 12
16 k0)2 16 l'/())2
(B20)
at I = 0. If the value of the expression shown in equation (B19) at I = 0 is
less than the
value of the expression shown in equation (B20) at I = 0, then si, will be
evaluated using
that equation and the expansion coefficients for si, are
+ RC)g14 )
¨ _____________________
csi ¨ (B21)
2
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c52 = 0 (B22)
1 ____________________
cs3 =¨V(R,õ + RC)gh, ) (B23)
8
c 0
54 ¨ ¨ (B24)
cs9 = 0 (B25)
with
sic') = -4, - fi¨V(Rõ +RC)gk ) . (B26)
[0098] Otherwise
1 / ___________________
c = Al(R + RC)g14 ) (B27)
81 4 w
c52 = --31/i(Rw + RC)g1,4 ) (B28)
4
cs3 = --1V(Rw + RC)gkr (B29)
16
cs4 = ¨3A/1(R + RC)gh(;) (B30)
16
cs9 = 0 (B31)
with
4 ) = (14, + ¨3 4R) = /I' + ¨1 V(Rw + RC)g14 ) ¨ ¨3 11(Rw + RC)gh(;) . (B32)
4 4 2 2
[0099] Similarly for SR, the following expressions
-4R = ii +11(Rw + RC)ghR =.4R = ii + V(Rw + RC)ghl()) +
Al(Rw + RC)g14) , 1 V(Rw + RC)g1,4 ) , (B33)
_________________ AR/ _______________ A2R / 2
214) 8 14)2
and
u* +11(Rw + RC)gh* = (-344, + -14 4R)= h + -32 V(Rw +Rc)gh1 ) - -21 V(Rw
+Roghp
3 V(Rw + RC)g14 ) , 1 V(Rw + RC)ghP ,
+ _______________ 4,1 _______________ AR/
4 1,4 ) 4 hP
3V(Rw+ROgli),1V(Rw + RC)gh(;) A2 ^
/2
R
16 k )2 I 16 hP2
(B34)
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are first evaluated. If the value of the expression shown in equation (B33) is
less than the
value of the expression shown in equation (B34), then SR will be evaluated
using that
equation and the expansion coefficients for 5R are
c5 =0 (B35)
cs6 = 11 (R + RC)gh(;)
(B36)
2
cs, =0 (B37)
cs8 = ¨1811 (Rs, + RC) gliP (B38)
C510 ¨ 0 (B39)
with
sT = -4õ - fi+ V(Rs, + RC)g111 ) . (B40)
[0100] Otherwise
c55 = ¨3V(R,, + RC)0(õ ) -- (B41)
4
1 i ______________________
Cs6 = --4Al (R,, + RC)ghP (B42)
cs, = --316Aii(R,, + RC)04 ) (B43)
1 _____________
c = ¨11(R,, + RC)ghP (B44)
s8 16
c = 0
510 (B45)
with
to,) 3 1 3 _________
S'R = (-41 -4R)= 11(Rõ + RC)g14
) ¨ 1 V (Rõ + RC)gh(;) . (B46)
4 4 2 2
[0101] Substituting equations (B1)-(B4) and equations (B15) and (B16)
into equation
(33), and retaining only terms of order o(1), o(AL), o(AR), o(A2, ), o(A2R),
o(ALAR), we
obtain
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= ________________________________________ =
s ¨FGL = /I' ¨ ¨FGR = Fi + sLsR (UR ¨
F*=Fi R
SR SL (B47)
= r()) ii*AL i +12*AR i +13*A2L i2 +14*A2R12 /5*ALAR12
where the coefficients are given by following equations
I") = asoeo (B48)
-*
= as0j1 as1j0 (B49)
7,*
J 2 = as0e2 as2e0 (B50)
= as0e3 asl as3e0 (B51)
7, *
J4 = as0e4 as2e2 as4e0 (B52)
7,*
J 5 = as0e5 as1e2 as2e1 as5e0 (B53)
[0102] In the above equations, the coefficients a0, as1, as2, as3, as4, as5
and the
coefficients eo, e1, e2, e3, e4, e5 are given by equations
as() = (sR(0) 40))-1 (B54)
(c, - csi)
as( = ___________________________________________________ (B55)
(4 ) -5,,(0))2
(cs6 cs2)
as2 = )(0)2
(B56)
(s(R - s)
(cs5 -c1)2 (cs7 -c3)
as3 = " 3 (0) 2
(0) (0) (B57)
(sR ) (sR - sL)
(cs6 -c2)2 (Cs8 -c4)
as4 = 3 (0) " (0) 2
(0) (B58)
(sR ) (sR - si, )
2(Cs6 Cs2)(Cs5 Cs1) (Cs10 Cs9 )
as5 = ____________________________________________________ (B59)
5R (0) -5,,(0))3 (sR(0) -5,,(0))2
(0)¨(0) (0)
Jo = (SR FGL SL FGR ) = n + do (B60)
= (c, FGL - cs FGR + sR"fi ) =ñ + (B61)
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e2 = (Cs6 ¨ Cs2(12 ) 4 ):1; (B62)
¨(o) ¨(o)
= (cs7FGL ¨ cs3FGR +C5/ + s(R U2) = + J3 (B63)
(o) _ (o) ¨
e4 = (Cs8 FGL ¨ s4 F GR s213 (L0) 14) = a 4 (B64)
z ¨0) ¨()
= csioFGL ¨ Cs9FG0R Cs6ji Cs1f3) = ñ J5 (B65)
where
= (Lo) SR(0) _ (7(Lo) )
(B66)
= (csis(Ro) cs540) )(UT) _ (7(L0)) _ is,(Lo)is, (R0 )
g1 (B67)
j2 = (cs2s(RO) cs6s(LO) )(U(RO) (7(LO) ) (L0)5 (RO)k2
(B68)
J3 = (c34 + cs74 ) cs1cs5)(U(R ) (4 )) (csi4 ) Cs54 )) (B69)
d4 = (c44 cs84 ) cs2cs6)(Ui? ) (4 )) k2(cs24 ) C64) (B70)
J5 = (c94 csiO4 ) cs2cs5 cs1cs6)((4 ) (4 ))
(cs24 ) Cs64 )) k2(cs14 ) Cs54 ))
(B71)
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APPENDIX C
[0103] Calculations of the source term related to the gravitational
forces.
[0104] The integration of the first term in the right-hand side of equation
(13) can be
written as
gdxdy = EJ SgdXdY
az
EL,EL,- (Rw + RC)gh ¨axdxdy
- (R,, + RC)gh ¨azdxdy
'
0
0
0
0
lEsõ,(L)L, _ (R,, + RC)ghdxdy
Es,v(i) _ (R,, + RC)ghdxdy
0
0
0
0
-s2 (R + RC)VQ,
E-S,v(1) (R + RC)VQ,
(Cl)
0
0
0
0
where 174 is the fluid volume on top of the triangle Q. . It can be calculated
as
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Vc2, = hdxdy
¨ z)dxdy
= + S(x ¨x)+ S(y ¨ ye)¨ ze ¨ Sri," (x ¨ xe)¨ Siiy(1)(y ¨ ye)_dxdy
(C2)
= ¨ zeidxdy + (S ¨ S(1))(x ¨ xe)dxdy + (S ¨ S(1))(y ¨ ye)dxdy
,;v rix
= ¨ ze)42, + ¨ ¨ .042, + ¨ ¨
[0105] Substituting Vfl, in equation (Cl) with equation (C2), the final
results of the
integration can be obtained.
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